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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.2.23a
Chapter 3, Problem 3.2.23a

21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4

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Start by recalling the definition of the derivative using limits: f'(x) = lim(h→0) [(f(x+h) - f(x))/h].
Substitute the given function f(x) = 4x² + 1 into the derivative definition: f'(x) = lim(h→0) [(4(x+h)² + 1 - (4x² + 1))/h].
Expand the expression (x+h)² to get x² + 2xh + h², and substitute it back into the limit: f'(x) = lim(h→0) [(4(x² + 2xh + h²) + 1 - 4x² - 1)/h].
Simplify the expression inside the limit: f'(x) = lim(h→0) [(4x² + 8xh + 4h² + 1 - 4x² - 1)/h] = lim(h→0) [(8xh + 4h²)/h].
Factor out h from the numerator and cancel it with the denominator: f'(x) = lim(h→0) [8x + 4h]. As h approaches 0, the term 4h vanishes, leaving f'(x) = 8x.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In mathematical terms, the derivative f'(x) is given by the limit: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.
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Derivatives

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is essential for defining derivatives and integrals. The limit allows us to analyze the function's behavior at points where it may not be explicitly defined, such as at points of discontinuity or at infinity.
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Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of finding derivatives, evaluating the function at points a=2 and a=4 is necessary to compute the derivative using the limit definition. This process helps in understanding how the function behaves at those specific points.
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Related Practice
Textbook Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

f(w) = w³ -w / w

Textbook Question

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>


a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)

Textbook Question

62–65. {Use of Tech} Graphing f and f'

a. Graph f with a graphing utility.

f(x)=e^−x tan^−1 x on [0,∞)

Textbook Question

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

a. (f^-1)'(4)

Textbook Question

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

a. Compute A'(t). What units are associated with this derivative and what does the derivative measure?

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Textbook Question

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?